Unfortunately Chinease New Year just started, so no action for the next two weeks at least 
LCSC seems to have an option to get an estimation:
(see “Check new stock and leadtimes”).
In parallel I asked for a quotation at shenzhen2u.com. Will be more expensive, but let’s see.
You may also use a local PCBA service such as Aisler in Germany. The advantage of all but JLCPCB is, that they order all components and put them on the board, after all arrived. JLCPCB is the only one I know, where you only get what is in stock at the moment of the order. Quite unconvenient, but they’re really cheap. The quality is ok as well, but that’s what I expect from everyone who does that job.
If you ask somewhere, please share this information here. Beautifying the BOM for a manufacturer always takes a lot of time, but imagine you would have to order everything yourself from different suppliers…